Question

12. If the sum of first 10 terms is 33 times the sum of first
5 terms of G.P., find the common ratio.​

Answer 1

• Sum of 10th term

S10 = a(1 - r¹⁰) / (1 - r)

• Sum of 5th term

S5 = a(1 - r⁵) / (1 - r)

The sum of 10th term is 33 times the sum of 5th term.

=> S10 = 33 × S5

a(1 - r¹⁰) / (1 - r) = 33 × a(1 - r⁵) / (1 - r)

a(1 - r¹⁰) = 33a × (1 - r⁵)

(1 - r¹⁰) = 33a/a × (1 - r⁵)

1² - (r⁵)² = 33 × (1 - r⁵)

(1 - r⁵) (1 + r⁵) = 33 × (1 - r⁵)

(1 - r⁵) (1 + r⁵) / (1 - r⁵) = 33

1 + r⁵ = 33

r⁵ = 33 - 1 = 32

r⁵ = 32 = 2⁵

•°• r = 2

The common ratio is 2.

Answer 2

let the first term of the G.P be a.

Let common ratio be r.

In a G.P sum of n terms = a(rⁿ - 1)/(r - 1)

Sum of first 10 terms in G.P = a(r¹⁰ - 1)/(r-1)

Sum of first 5 terms in G.P = a(r⁵-1)/(r - 1)

Given,

sum of first 10 terms of G.P = 33 x (sum of first 5 terms of G.P)

a(r¹⁰ - 1)/(r - 1) = a(r⁵ - 1)/(r - 1)

⇒ r¹⁰ - 1 = 33 (r⁵ - 1)

⇒ (r⁵)² - 1² = 33 (r⁵ - 1)

⇒(r⁵ - 1)(r⁵ + 1) = 33(r⁵ - 1) [a² - b² = (a+b)(a-b) ]

⇒ r⁵ + 1 = 33

⇒ r⁵ = 32

⇒ r = 2

Common ratio of G.P = 2